We saw in the tutorial about Summing Amplifiers that the voltages or signals applied to the multiple inputs of an inverting operational amplifier circuit can be “summed” together to produce a single output, and depending on the amplifiers configuration, inverting or non-inverting, the output signal will be a positive or negative sum of all its inputs.

We also saw that the summing amplifier multiplies each input voltage by its weighted gain determined by the ratio of R_{ƒ}/R_{IN}, that is the ratio of the feedback resistor (R_{ƒ}) to the corresponding input resistor, (R_{IN}).

The summed output voltage (or signal) could be the result of the direct addition method, in which each input resistor (R_{IN(1)} to R_{IN(n)}) have the same values producing a linear output voltage corresponding to these values, or it could be the result of the binary-weighted method in which each input resistor is doubled in value producing a stepped output voltage which corresponds to the “weight” of each input value. Summing amplifiers have many electronics applications, such as in audio mixer designs or in analogue-to-digital conversion (ADC), etc.

But as well as using operational amplifiers as *summing amplifiers* (addition) or as *differential amplifiers* (subtraction), we can also configure multiple input opeartional amplifier circuits to function as an *Averager* circuit which can produce an output voltage that corresponds to the average voltage value of two or more inputs.

## Passive Averager

The **Passive Averager** is basically a resistive network or circuit configured to provide an output voltage whose value is equal to the mathematical *average* of all of its input voltages. Any number of inputs can be used to form an averager circuit, either passive or active. Consider the 2-input resistive circuit below.

Here the two resistors, R_{1} and R_{2} are connected together so that one end of each resistor forms a common junction or node, while a voltage source is applied to the other end of each resistor as shown.

This then forms the basis of a *passive averager circuit* which produces an output voltage equal to the average value of the two input voltages as they are effectively connected together through the resistors. This basic circuit configuration can also be used for summing and subtractor circuits.

Kirchhoff’s current law (KCL) states that the algebraic sum of all the electrical currents entering and leaving a circuit junction or node must must be equal to zero. So the sum of the currents passing through this passive resistive circuit will be equal to: I_{T} = I_{R1} + I_{R2}.

Therefore:

This basically means that V_{OUT} is equal to the sum of the inputs currents divided by the reciprocal value of the individual resistors, as the resistors are effectively connected together in parallel through the voltage sources, and this idea forms part of Millman’s Theorem. That is V = I/G, where “G” is conductance. Then we can expand this basic 2-input passive averager equation for resistive circuits with multiple inputs of 3, 4 or more resistors and voltages as shown.

### Passive Averager Equation

Thus any number of inputs may be used to produce a passive averager circuit with the voltage seen at the common node being the mathematical average of all the input voltages.

## Passive Averager Example No1

A 2-input passive averager circuit is constructed using a 2kΩ and a 4kΩ resistor connected together. If a voltage supply of 12 volts d.c. is connected to one end of the 2kΩ resistor and a second voltage source of 6 volts d.c. is connected to one end of the 4kΩ resistor. Calculate the output voltage at the common junction.

Firstly assume: R_{1} = 2kΩ, R_{2} = 4kΩ, V_{1} = 12V, and V_{2} = 6V.

So the common node junction voltage was calculated as 10 volts. But you may be sat there thinking that: (12 + 6)/2 = 9 volts. The average voltage output should be 9 volts, and you would be correct. However, the two resistors used in this example are of different values, 2kΩ and 4kΩ, so will influence the currents flowing through the resistive network producing what is known as a *Weighted Averager Circuit*. That is each input is multiplied by its weighting factor before being averaged.

In fact for this simple example, I_{R1} will be: (12-10)/2000 = +1mA flowing into the junction, and I_{R2} will be: (6-10)/4000 = -1mA flowing out of the junction. That is 1mA of current is flowing from the larger 12 volt supply to the smaller 6 volt supply through the common junction.

However, if we made these two input resistances of equal value so that: R_{1} = R_{2} = R, then the current flowing through the junction would be zero as the two currents I_{R1} and I_{R2} are the same but opposite value so cancel. Then the passive averager equation above would simplify down too:

### Passive Averager Equation

That is with equal resistance values instead of different individual resistance values, the output voltage value at the common junction will be exactly equal to the average value of the individual voltage sources, making it a true passive averager circuit. Then using our simple 2-input averager circuit above, V_{OUT} = (V_{1} + V_{2})/2 = (12 + 6)/2 = 9 volts, as we would expect.

## Passive Averager Example No2

A 4-input passive averager circuit is constructed using the following resistive values: R_{1} = 4KΩ, R_{2} = 11KΩ, R_{3} = 20KΩ, and R_{4} = 30KΩ. If the corresponding voltages applied to these resistances are: V_{1} = 20V, V_{2} = 15V, V_{3} = 45V, and V_{4} = 60V. Calculate the passive resistive networks output voltage, and again with ALL the resistances being equal in value.

With all resistor values being of equal value and represented as “R”

We can see that the values of the individually connected resistors make a big difference to the value of the output voltage, V_{OUT} as the weighted average value was calculated as 25 volts, whereas the true average voltage value was calculated as 35 volts.

Both examples where included here as the fist method forms the basis of *Millman’s Theorem* in which any number of parallel resistive and voltage branches can be reduced to one single value, and for our simple example, the four voltage sources produced a single output voltage of 25 volts.

## Op-amp Averager Circuit

One main disadvantage of the passive averager circuit above, is that its output voltage can be influenced by a connected load, especially if the load is of a low impedance. But we can ensure that the average output voltage of the *passive averager circuit* remains true and constant by converting it into an active averager circuit with the addition of an operational amplifier on its output.

The simplest and easiest way to do this is to connect the output of the resistive averager network to the input of an operational amplifier, or “op-amp”, configured as a non-inverting “voltage follower”. A voltage follower is basically a unity-gain buffer which produces a positive output voltage as shown.

### Averager Circuit Using a Voltage Follower

As we saw in previous tutorials, the input impedance of an op-amp is extremely high, so no current flows into the non-inverting input terminal. As the op-amps output is connected directly back to its inverting input, the feedback will therefore be 100%, so V_{IN} is exactly equal to V_{OUT} giving the op-amp a fixed gain of “1” or unity.

That is V_{OUT} = V_{IN} producing a positive output averager circuit. The advantage here is that the individual inputs are effectively isolated from each other, and therefore any connected load, so any number of inputs may be used.

## Inverting Averager

We can also configure the operational amplifier as an inverting amplifier to produce a negative output average voltage. The closed-loop voltage gain (A_{V(CL)}) due to the feedback path between the output and input terminals is given as:

A_{V(CL)} = -R_{ƒ} / R_{IN} = V_{OUT} / V_{IN}

Then we can re-write this as being:

But for our averaging amplfier, V_{IN} = V_{1} + V_{2} + V_{3} + … + etc. So if for simplicity, we use a 3-input averager circuit, then the expression for the output voltage becomes:

Thus each input voltage is multiplied by a common factor of -R_{ƒ}/R_{IN}. If we make all the resistance values equal and the same, that is the feedback resistor R_{ƒ} = R_{IN} = “R”, and the number of inputs equal to 3. Then R_{ƒ} = R_{IN1} = R_{IN2} = R_{IN3} = R, and “n” = 3, then the above equation becomes:

Setting the closed-loop voltage gain of the operational amplifier equal to the reciprocal value of the number of inputs, which is 3 in this given example, the output voltage from the op-amp averager circuit will be inverted (-V_{OUT}), and the mathematical average value of the three individual inputs as shown.

### Inverting Averager Circuit

In this simple example of an inverting op-amp **Averager Circuit** we have used 3-inputs, but the circuit can be configured to use any number of inputs providing that ALL the input resistances are set to equal the value of *n*R*, where “R” is the resistive value of the feedback resistor, “n” is the number of input channels, and “n*R” is the resistive value of all the individual input resistors, otherwise the averaging amplifier becomes a summing amplifier.